Sampling from a log-concave distribution with Projected Langevin Monte Carlo

TL;DR

This paper introduces a projected Langevin Monte Carlo algorithm to efficiently sample from log-concave distributions with compact support, providing polynomial-time guarantees and empirical performance comparable to existing methods.

Contribution

The paper extends Langevin Monte Carlo with a projection step for compact supports, establishing polynomial mixing time bounds and demonstrating competitive empirical results.

Findings

LMC mixes in $ ilde{O}(n^7)$ steps for uniform distributions.

Preliminary experiments show LMC performs comparably to hit-and-run.

LMC provides a new Markov chain approach for sampling from log-concave measures.

Abstract

We extend the Langevin Monte Carlo (LMC) algorithm to compactly supported measures via a projection step, akin to projected Stochastic Gradient Descent (SGD). We show that (projected) LMC allows to sample in polynomial time from a log-concave distribution with smooth potential. This gives a new Markov chain to sample from a log-concave distribution. Our main result shows in particular that when the target distribution is uniform, LMC mixes in $\tilde{O}(n^7)$ steps (where $n$ is the dimension). We also provide preliminary experimental evidence that LMC performs at least as well as hit-and-run, for which a better mixing time of $\tilde{O}(n^4)$ was proved by Lov{\'a}sz and Vempala.